Questions tagged [monomorphisms]

For questions related to monomorphisms, which are categorical generalizations of injective functions.

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Find a monomorphism function $U_7\rightarrow S_5$

Find a monomorphism $f$ function $U_7\rightarrow S_5$. I know that: $\phi(7)=6$ $|S_5|=120$ I tried to assign permutations but I'm not sure how: $f(1)=id$ $f(2)=(1 2)$ $f(3)=(3 4)$ $f(4)=f(2)f(2)$ $f(...
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1answer
57 views

Totally decomposable vectors as quotient of monomorphisms by SL

Consider vector spaces $V$ and $W$ of dimensions $m$ and $n\geq m$, respectively. Both the special linear group $\operatorname{SL}(V)$ as well as the general linear group $\operatorname{GL}(V)$ act ...
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1answer
45 views

a map in $R$-Mod is injective iff it is a monomorphism [closed]

I would like to see a detailed argument of this fact: a map in $R$-Mod is injective iff it is a monomorphism
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1answer
40 views

How do I see that the map $X^n \to X^{n+1}$ of CW-building blocks is an embedding?

In his book “A concise course in algebraic topology”, May defines a CW complex inductively as being the union of increasing subspaces $X^n$, where $X^0$ is discrete $X^{n+1}$ is the simultaneous ...
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1answer
36 views

How to prove that there is not a monomorphism from Klein 4-group to $Z_6$(or a epimorphism from $Z_6$ to $V_4$)?

How to prove that there is not a monomorphism from Klein 4-group to $Z_6$(or a epimorphism from $Z_6$ to $V_4$) ? I know that: If $f$ is a monomorphism than $Kernel(f)={0}$ . In $V_4$ , three ...
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2answers
58 views

Do I need an “explicit” identity morphism to have a category?

I'm trying to study category theory as an autodidact, using Bartosz Milewski's Category Theory for Programmers, and I've just finished chapter 1, where the last question/exercises is whether a ...
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1answer
47 views

Conditions for adjoints to preserve monos?

Given an adjoint pair $L \dashv R$ and a mono $f \in \text{Hom}(X, RY)$, what are some conditions which will guarantee $\tilde{f} \in \text{Hom}(LX, Y)$ is still a mono? Obviously this becomes easy if ...
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1answer
51 views

Monomorphisms Between Infinite Graphs

I'm really curious about this question: Let $G(V_G,E_G)$ and $H(V_H,E_H)$ be infinite (infinite!, not finite) graphs, such that $$|V_G|=|V_H|,$$ and let $f$, $g$ be functions $f: G\rightarrow H$, $g: ...
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2answers
27 views

Examples of homomorphisms

Could someone give examples homomorphisms of rings f: R->S and g: S->T such that gof is a monomorphism but g is not? I tried with the maps from Z (a->na) but can't think of a map such that ...
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54 views

Function-like monos and epis

When dealing with $\textbf{Set}$ we have that if $f:A\to B$ is a monomorphism, $g:A\to A’$ is an epimorphism, and adding $f’:A’\to B$ we have a commuting triangle, then $f’$ must be a monomorphism. ...
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0answers
66 views

Show that two direct limits (infinitely generated Abelian groups) are not isomorphic

I have a process which generates infinite sequences $S=S_0\rightarrowtail S_1 \rightarrowtail S_2\ldots$ of finite Abelian groups $S_i$ connected by monomorphisms $S_i\rightarrowtail S_{i+1}$ (...
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0answers
90 views

The Cantor-Bernstein Theorem in Categories

The Cantor-Bernstein theorem states that if there are two injective applications $f:A\to B$ and $g:B\to A$ between sets $A$ and $B$, then there exists a bijection $h:A\to B$. I asked myself the ...
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1answer
49 views

Fibrations are thought of as epimorphisms

In the book More concise algebraic topology on the page 213 they write We think of fibrations as analogous to epimorphisms. BUT Hovey on the page 51 says $f$ is a fibration if it is in $J-inj$. My ...
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2answers
135 views

Monomorphisms and epimorphisms in the category of chain complexes

Let $\mathsf{C}$ be an abelian category and $\mathsf{Comp(C)}$ its category of chain complexes. Suppose that $f\colon (C,d)\to (C',d')$ is a monomorphism in $\mathsf{Comp(C)}$. I want to prove that ...
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0answers
47 views

Can monomorphisms between two objects imply isomorphism?

I have a very basic understanding regarding category theory, but I was wondering whether there is a condition for isomorphism between objects via monomorphisms? More concretely, given a category $\...
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1answer
37 views

Is the image a universal object?

Given a function $f:X\to Y$ in category $\mathcal{C}$, one can construct the image as a factorisation $f=(e:I\hookrightarrow Y)\circ(g:X\to I)$ that is universal (initial) among all such ...
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1answer
135 views

In Set, why are projections not epic but injections are monic?

I'm working through Bird and DeMoor's Algebra of Programming and I have some basic gaps in my understanding. Problem 2.28 asks if projection outl is epic in Set, if inl is monic, and why the ...
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1answer
56 views

Relation between monomorphicity and faithfulness?

I'm a beginner in cat theory, so this may seem straight-forward, but I didn't manage to untangle anything by trying to write it calmly... This question arised from the fact that in Cat, whose ...
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1answer
53 views

Modules: monos are stable under pushouts

In $R$-Mod, monos are stable under pushouts: suppose in $R$-Mod that $f_1:M \rightarrowtail M_1$ is a mono and $f_2:M\to M_2$ so that they form a span. Complete this to a pushout $\hat{f}_2:M_1\to N$ ...
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0answers
108 views

Immersion of schemes is a monomorphism

Any immersion in the category of schemes is a monomorphism. Now, I now that $f: X \rightarrow Y$ is a monomorphism if and only if $\Delta_f: X \rightarrow X \times_Y X$ is an isomorphism. To prove ...
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1answer
39 views

Find a ring embedding from a finite field to a Galois ring

Let $\mathbb{F}_q$ be a finite field with $q$ elements and $GR(p^s,r)$ be a Galois ring with $p^{sr}$ elements and chacteristic $p^s$. I know that $$GR(p^s,r)/pGR(p^s,r)\cong \mathbb{F}_{p^r}.$$ A ...
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1answer
107 views

Monomorphism & epimorphism in the category of schemes

Is there a morphism in the category of schemes which is simultaneously a monomorphism and an epimorphism yet is not an isomorphism? "Nicer" examples are preferred (e.g. with integral Noetherian ...
2
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1answer
67 views

Bimorphism which is neither injective nor surjective

In Ring the inclusion $f:\mathbb Z \to \mathbb Q$ is a not surjective but injective bimorphism. In Div the quotient map $g:\mathbb Q \to \mathbb Q / \mathbb Z$ is a not injective but surjective ...
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2answers
128 views

Why is $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ always injective?

Let $R$ be a commutative ring with $1$. For all $R$-modules $V,W$ we have a canonical $R$-linear map $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ from tensor product of dual modules ...
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1answer
174 views

Monomorphisms preserved in a pullback

The question is answered, e.g, here. Suppose $(P, p_1, p_2)$ is a pullback for $f:A\to C$ and $g:B\to C$ with $fp_1 = gp_2$. if $f$ is a monomorphism, show $p_2$ is a monomorphism. What we do is ...
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4answers
428 views

Counterexample: functor does not preserve monomorphisms

I need to show that functors need not preserves mono's and epi's. For epi's, I have as counterexample the forgetful functor $F : \mathbf{Ring} \to \mathbf{Set}$. We have that $f: \mathbb{Z} \...
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0answers
84 views

In R-Mod, all monomorphisms are equalizers

I want to prove that in the category of R-Modules, all monomorphisms are equalizers. We start by assuming that if $f : A \to B$ is mono and if $f$ equalizes $g : B \to C$ and $h : B \to C$ (i.e. $ g \...
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1answer
140 views

Are all split epimorphisms effective? [duplicate]

An epimorphism is called split if it has a section (a right inverse). An epimorphism is called effective if it has a kernel pair and is the coequalizer of its kernel pair. The ncatlab implies here ...
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1answer
68 views

Every $\lambda-$pure morphism in a locally $\lambda-$presentable category is a regular monomorphism

Consider the page from the book by Adamek & Rosicky: Locally presentable and accessible categories. given below. I need to derive this: Here is the statement (2) to prove the universal ...
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1answer
73 views

Proof verification: An arrow which is monic under a faithful functor is itself monic

Context: To introduce some symbols and such, what I'm seeking to prove is this: Let $F$ be a faithful functor. Suppose $F(f)$ is a monic arrow. Show $f$ is monic. This came up as part of a class ...
2
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1answer
30 views

Investigate whether the given transformation is a monomorphism / epimorphism. Find image and kernel

I have serious doubts - I will be very grateful if someone will help me here Investigate whether the given transformation is a monomorphism / epimorphism. Find its image and kernel. $$ F \in L(\...
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1answer
101 views

$\lambda$-pure morphisms in $\lambda$-accessible categories are monos, unclear proof

This is Proposition 2.29 from the book Locally Presentable and Accessible Categories by Jiří Adámek and Jiří Rosický. Above is a proof that $\lambda$-pure morphisms in $\lambda$-accessible categories ...
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1answer
274 views

Morphism=monomorphism•epimorphism?

Is it true that any morphism in any category can be written as a combination of monomorphism and epimorphism? In SET and categories where monomorphism is an injective function and epimorphism is a ...
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3answers
1k views

Epimorphism and monomorphism explained without math?

I'm trying to understand category theory to increase my coding skills and epimorphism and monomorphism aren't clear to me. Unfortunately, my last formal education was when I was 12 due to ...
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1answer
49 views

Is the unique subobject also mono?

Suppose $u : S → A$ and $v : T → A$ with codomain $A$ are monomorphisms, we write $u ≤ v$ if $u$ factors through $v$—that is, if there exists $φ : S → T$ such that $u = v ∘ φ$. I understand that ...
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1answer
22 views

Relation between homomorphisms and monomorphims of finite groups

For any finite groups L and G, let $h(L,G)$ denote the number of homomorphisms from L to G and $i(L,G)$ denote the number of monomorphisms from L to G. Proof that $$h(L,G)=\sum_{N \triangleleft \ L} ...
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1answer
87 views

Existence of morphisms in a free completion under directed colimits,$\lambda$-accessible category

Let $\cal K$ be a $\lambda$-accessible category with directed colimits and $\cal C$ be its representative full subcategory consiting of $\lambda$-presentable objects. Let $\cal L$ be free completion ...
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1answer
35 views

Monomorphisms, unclear basic property, Functor

Suppose that for morphisms in a category it holds that $f\circ u=v\circ f'$ and $g\circ u=v\circ g'$ and that for a functor $F$, $Fv$ is a monomorphism. Suppose that $Ff'$ and $Fg'$ are distinct. WHY ...
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0answers
96 views

A question about linearly independent monomorphisms.

My question was too big to put it in the title: Having a set of linearly independent field monomorphisms $K→L$, is it still possible to express one of these monomorphisms as a linear combination of ...
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2answers
501 views

How exactly is a Field monomorphism defined?

Multiple times in my question I'll use the term 'a mapping of $x$', what I mean with this is one of the element to which $x$ can be mapped to (since there could be more than one). We have Two fields, ...
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3answers
277 views

Is an injective function always monomorphic

Is an injective function monomorphic (in category of sets)? How do I prove this? Conversely, I could prove that a monomorphic function is injective in the following manner: If f is monic, then $f\...
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2answers
79 views

Is there a counter example for monmorphism

Monomorhism as defined is: A morphism $f: A \to B$ is a monomorphism if for every object $C$ and every pair of morphisms $g,h: C \to A$ the condition $f\circ g = f\circ h$ implies $g = h$. Applying ...
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1answer
47 views

Plain monomorphism,the defintion

In several places the notions of a plain monomorphism or a plain epimorphism are used, but never defined. See eg. here on page 60, or search google with query plain monomorphisms tholen Do they ...
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2answers
93 views

Confused about the definition of a Field Extension

A particular author defines a Field Extension as a monomorphism (to be more detailed, as an injective homomorphism) between two Fields. However, my idea of a Field Extension is that of a pair of ...
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1answer
149 views

A counterexample to Cantor-Schroeder-Bernstein in groups.

We need to find a counterexample to the Cantor-Schroeder-Bernstein theorem for the category of groups (that is, find two monomorphisms $\varphi\colon G\to H$ and $\psi\colon H\to G$ such that $G$ and $...
4
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0answers
163 views

Mono(epi)morphisms as im(sub)mersions in the category of manifolds

In the definition of the category of smooth manifolds it's usual to take the morphisms to be simply smooth functions, and this results in having monomorfisms and epimorphisms to be injective and ...
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3answers
331 views

How do kernels prove how a function fails to be injective

I know that for $f\colon X \to Y$, where $e_Y$ is the identity of $Y$: $$ \ker(f) = \left\{x \in X \, \middle| \, f(x) = e_Y \right\} $$ I've learnt that kernels imply how much a homomorphism fails ...
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2answers
87 views

Is $\mathbb Z_7$ an extension of $\mathbb Z_2$?

Is $\mathbb Z_7$ a extension of $\mathbb Z_2?$ I know it is not, because we can't establish a monomorphism between $\mathbb Z_7$ and $\mathbb Z_2$,i.e. it doesn't exist an injective homomorphism ...
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2answers
995 views

Any category with zero object, every kernel is monomorphism.

This is the problem 5.53 of Rotman, Homological Algebra. In any category having a zero object calling it $0$, prove that every kernel is monomorphism. Let $f:A\to B$ and $\ker(f):K\to A$ be the ...
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1answer
180 views

Prove linear transformation is monomorphism

I have troubles with following exercise: $(X, \langle \cdot, \cdot \rangle_X)$ and $(Y, \langle \cdot, \cdot \rangle_Y)$ are euclidean spaces. Linear transformation $f \in L(X, Y)$ has property $$\...